Math practice · reimagined
Linear Functions Game
In this linear functions fight, every slope between two points is a whole number from −4 to 4, never 0, and every system pairs x + y with x − y. Your rival gets the same points, so read the line faster and your fighter hits first.
Ages 4–99+ · no download · plays in your browser · 23 languages
What is a linear function?
A linear function is a rule where y changes by the same amount each time x grows by 1, so its graph is a straight line. It is written y = mx + b: m is the slope, or steepness, and b is where the line crosses the y-axis. A 20 cm candle burning 2 cm an hour follows y = 20 − 2x.
Example: x + y = 17 and x − y = 9
- The game gives you two equations at once and asks only for x: x + y = 17 and x − y = 9.
- Add the two equations. The +y and the −y cancel, leaving 2x on the left and 17 + 9 = 26 on the right.
- If 2x = 26, then x is half of 26, so x = 13.
- Check with y = 17 − 13 = 4: since 13 − 4 = 9, both equations hold, and you type 13.
Learn it step by step
Slope is how steep a line is, and it is measured as rise over run: how far the line climbs for every step it takes sideways.
Given two points, subtract the y-values to get the rise and the x-values to get the run, then divide. From (1, 2) to (4, 11): the rise is 11 − 2 = 9, the run is 4 − 1 = 3, so the slope is 3. The line goes up three for every one across.
Order matters only in that you must be consistent — start at the same point for both subtractions, or you will get the sign backwards.
A positive slope climbs to the right and a negative one falls. The number is a rate: on a distance-time graph it is speed, on a cost graph it is price per item.
Find the rise, find the run, divide, and type the slope.
Tips and common mistakes
- Look at the signs in front of y. Opposite signs, like +y and −y, cancel when you add the equations; matching signs cancel when you subtract one from the other.
- A common slip is stopping at 2x = 30 and typing 30. Adding the equations gives twice x, so halve it before you answer: x = 15.
- For straight line graphs practice at home, plot y = 2x + 1 on squared paper from x = 0 to 3. Each point sits 2 squares higher, and a ruler joins them.
How to play
The slope level shows two points, such as slope from (2, 5) to (4, 3), and you type −1, with the minus key for falling lines. Level two is a systems of equations game: from x + y = 22 and x − y = 8 you type only x, 15. Answer before your rival and Mushroom strikes, harder the faster you are.
— Quick answers —
Linear functions: the short FAQ
- What is a system of linear equations?
- A system of linear equations is a set of equations that share the same unknowns and must all be true at once. Alone, x + y = 12 has endless solutions: 5 and 7, 9 and 3, and many more. Add x − y = 2, and only x = 7, y = 5 fits both. On a graph, that pair is the point where the two lines cross.
- How do you solve a system of equations by elimination?
- To solve by elimination, make one letter cancel, then solve for the other. In 3x + 2y = 16 and x − y = 2, the y terms do not cancel yet. Multiply the second equation by 2 to get 2x − 2y = 4, then add the two: 5x = 20, so x = 4. Putting 4 back into x − y = 2 gives y = 2.
- How do you remember which difference goes on top in the slope?
- Test the formula on a line you already know. The line y = 2x passes through (0, 0) and (1, 2), so it climbs 2 for every 1 across, and its slope must be 2. That only comes out as 2 over 1, never 1 over 2, so the change in y always goes on top and the change in x underneath.
- Why does my child think a negative slope is less steep?
- Many learners read −4 as a gentler slope than 2, because −4 is the smaller number. Steepness, though, is the size of the slope with its sign ignored: −4 drops 4 for every step across, while 2 climbs only 2. Compare the sizes to judge steepness, and use the sign only to tell whether the line rises or falls.
- Can a teacher use this game on linear functions with a class for free?
- Yes, a teacher can: both levels are free and run in the browser that school devices already have, with no sign-ups or downloads. For slope practice at the board, draw a grid, mark (0, 1) and ask pupils to add points whose slope from it is 2. Answers such as (2, 5) and (−1, −1) all land on one straight line. On the schools plan, billed per pupil, a teacher dashboard shows who is getting steadier with slopes.